nLab fiber integration in ordinary differential cohomology

Idea

The special case of fiber integration in differential cohomology for ordinary differential cohomology is the partial higher holonomy operation for circle n-bundles with connection:

for $Y \to X$ a bundle of compact smooth manifolds $S$ of dimension $k$ and $[\nabla] \in H_{diff}^n(Y)$ a class in ordinary differential cohomology of degree $n$ on $Y$, its fiber integration

$$\left[\exp(i \int_{Y/X} \nabla)\right] \in H^{n-k}_{diff}(X)$$

is a differential cohomology class on $X$ of degree $k$ less.

In the particular case that $X = *$ is the point and $dim Y = k = n-1$ the element

$$\exp(i \int_{Y} \nabla) \in H^{1}_{diff}(*) \simeq U(1)$$

is the higher holonomy of $\nabla$ over $Y$.

Definition

Differential orientation

The operation of fiber integration in generalized (Eilenberg-Steenrod) cohomology requires a choice of orientation in generalized cohomology. For fiber integration in differential cohomology this is to be refined to a differential orientation .

Accordingly, instead of a Thom class there is a differential Thom class .

Definition

For $X$ a compact smooth manifold and $V \to X$ a smooth real vector bundle of rank $k$ a differential Thom cocycle on $V$ is

• a compactly supported cocycle $\hat \omega$ in the ordinary differential cohomology of degree $k$ of $V$;

• such that for each $x \in X$ we have

  $$\int_{V_x} \omega = \pm 1 \,.$$

Remark

The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ in compactly supported integral cohomology is an ordinary Thom class for $V$.

Definition

Let $p : X \to Y$ be a smooth function of smooth manifolds.

An $H \mathbb{Z}_{diff}$-orientation on $p$ is

1. A factorization through an embedding of smooth manifolds

   $$p : X \hookrightarrow Y \times \mathbb{R}^N \stackrel{}{\to} Y$$

   for some $N \in \mathbb{N}$;

2. a tubular neighbourhood $W \hookrightarrow Y \times \mathbb{R}^N$ of $X$;

3. a differential Thom cocycle, def. , $U$ on $W \to X$.

This appears as (HopkinsSinger, def. 2.9).

Via differential Thom cocycles

Write $H^n_{diff}(-)$ for ordinary differential cohomology. For any choice of presentation, there is a fairly evident fiber integration of compactly supported cocycles along trivial Cartesian space bundles $Y \times \mathbb{R}^N \to Y$ over a compact $Y$:

$$\int_{\mathbb{R}^N} : H^{n+N}_{diff,cpt}(Y \times \mathbb{R}^n) \to H^n_{diff}(Y) \,.$$

Definition

Let $X \to Y$ be a smooth function equipped with differential $H\mathbb{Z}$-orientation $U$, def. . Then the corresponding fiber integration of ordinary differential cohomology is the composite

$$\int_{X/Y} : H_{diff}^{n+k}(X) \stackrel{(-)\cup U}{\to} H_{diff, cpt}^{n+N}(X \times \mathbb{R}^N) \stackrel{\int_{\mathbb{R}^N}}{\to} H_{diff}^n(Y) \,.$$

This appears as (HopkinsSinger, def. 3.11).

In terms of Deligne cocycles

We discuss an explicit formula for fiber integration along product-bundles with compact fibers in terms of Deligne complex, following (Gomi-Terashima 00).

For $X$ a smooth manifold, write $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ for the Deligne complex in degree $(n+1)$ over $X$.

Definition

Let $X$ be a paracompact smooth manifold and let $F$ be a compact smooth manifold of dimension $k$ without boundary. Then there is a morphism

$$\int_F \;\colon\; \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \to \mathbf{H}(X, \mathbf{B}^{n-k} U(1)_{conn})$$

given by (…)

(Gomi-Terashima 00, section 2, corollary 3.2)

In terms of smooth homotopy types

The above formulation of fiber integration in ordinary differential cohomology serves as a presentation for a more abstract construction in smooth homotopy theory.

Let $\mathbf{H} \coloneqq$ Smooth∞Grpd be the ambient cohesive (∞,1)-topos of smooth ∞-groupoids/smooth ∞-stacks. As discussed there, the Deligne complex, being a sheaf of chain complexes of abelian groups, presents under the Dold-Kan correspondence a simplicial presheaf on the site CartSp, which in turn presents an object

$$\mathbf{B}^n U(1)_{conn} \in \mathbf{H} \,,$$

discussed here: the smooth moduli ∞-stack of circle n-bundles with connection.

Let now $\Sigma_k$ be a compact smooth manifold of dimension $k \in \mathbb{N}$ without boundary. There is the internal hom in an (infinity,1)-topos

$$[\Sigma_k, \mathbf{B}^n U(1)_{conn}] \in \mathbf{H} \,,$$

which is the smooth moduli $n$-stack of circle $n$-connections on $\Sigma_k$.

Proposition

For all $k \leq n$ there is a natural morphism

$$\exp(2\pi i\int_\Sigma(-)) \; \colon \; [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k} U(1)_{conn} \;\;\; \in \mathbf{H} \,.$$

which for $U \in$ SmthMfd a smooth test manifold sends $n$-connections on $\Sigma_k$ on $U \times \Sigma_k$ to the $(n-k)$-connection on $U$ which is their fiber integration over $\Sigma_k$.

(Fiorenza-Sati-Schreiber 12)

Proof

To see this, observe that

1. by definition $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;

2. if $\{U_i \to \Sigma_k\}$ is a fixed good open cover of $\Sigma_k$, then $\{U \times U_i \to U \times \Sigma_k\}$ is also a good open cover, for every $U \in$ CartSp;

3. hence the Cech nerve $C(\{U \times U_i\})$ is a natural (functorial in $U \in CartSp$) cofibrant object resolution of $U \times \Sigma_k$ in the projective local model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$ which presents $\mathbf{H} =$Smooth∞Grpd (as discussed there);

4. the (image under the Dold-Kan correspondence) of the Deligne complex $\mathbb{Z}(n+1)^\infty_D$ is a is fibrant in this model structure (since every circle $n$-bundle is trivializable over a contractible space $U \in$ CartSp).

This means that a presentation of $[\Sigma_k, \mathbf{B}^n U(1)_{conn}]$ by an object of $[CartSp^{op}, sSet]_{proj,loc}$ is given by the simplicial presheaf

$$U \mapsto DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))$$

that sends $U$ to the Cech-Deligne hypercohomology chain complex with respect to the cover $\{U \times U_i \to U \times \Sigma_k\}$.

On this def. provides a morphism of simplicial sets

$$DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\})) \to DK \mathbb{Z}(n+1)^\infty_D(U)$$

which one directly sees is natural in $U$, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in $\mathbf{H}$.

Applications are to

• transgression

• double dimensional reduction

Properties

General

(…)

Abstract formulation

At least the fiber integration all the way to the point exists on general grounds for the intrinsic differential cohomology in any cohesive (∞,1)-topos: the general abstract formulation is in the section Higher holonomy and Chern-Simons functional and the implementation in smooth ∞-groupoids is in the section Smooth higher holonomy and Chern-Simons functional .

Examples

$\infty$-Chern-Simons functionals in higher codimension

(…)

Differential universal characteristic class / extended $\infty$-Chern-Simons Lagrangian:

$$\hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^{n}U(1)_{conn}$$

moduli $\infty$-stack of higher gauge fields on a given $\Sigma_k$:

$$[\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}$$

Lagrangian of $\hat \mathbf{c}$-Chern-Simons theory:

$$[\Sigma_k, \hat \mathbf{c}] : [\Sigma_k, \mathbf{B}G_{conn}] \to [\Sigma_k, \mathbf{B}^n U(1)_{conn}]$$

extended action functional of $\hat \mathbf{c}$-Chern-Simons theory in codimension $(n-k)$

$$\exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \hat \mathbf{c}] ) : [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i\int_{\Sigma_k} (-))}{\to} \mathbf{B}^{n-k} U(1)_{conn} \,.$$

(…)

Related concepts

• higher parallel transport, higher holonomy

• fiber integration in differential K-theory

References

A discussion in the general sense of fiber integration in generalized (Eilenberg-Steenrod) cohomology is in section 3.4 of

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory

and around prop. 2.1 (in the context of Chern-Simons theory) in

• Daniel Freed, Classical Chern-Simons theory II. Special issue for S. S. Chern. Houston J. Math. 28 (2002), no. 2, 293–310.

Explicit formulas for fiber integration of cocycles in Cech-Deligne cohomology are given in

• Kiyonori Gomi and Yuji Terashima, A Fiber Integration Formula for the Smooth Deligne Cohomology International Mathematics Research Notices 2000, No. 13 (pdf, pdf)

and their generalization from higher holonomy to higher parallel transport in

• Kiyonori Gomi and Yuji Terashima, Higher dimensional parallel transport Mathematical Research Letters 8, 25–33 (2001) (pdf)

and

• David Lipsky, Cocycle constructions for topological field theories (2010) (pdf)

See also

• Johan Dupont, Rune Ljungmann, Integration of simplicial forms and Deligne cohomology Math. Scand. 97 (2005), 11–39 (pdf)

The observation that the construction in Gomi-Terashima 00 induces refines to smooth higher moduli stacks is discussed in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Extended higher cup-product Chern-Simons theories, Journal of Geometry and Physics Volume 74, 2013, Pages 130–163 (arXiv:1207.5449)

for the case without boundary and for the general case in

• Urs Schreiber et al. Local prequantum field theory.